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AD-A265 214
WL-TR-93-3002
CALCULATIONS ON UNSTEADY TYPE IV INTERACTION ATMACH 8
Datta GaitondeUniversal Energy Systems4401 Dayton-Xenia RoadDayton, OH 45432
January 1993
Final Report for Period December 1991 - August 1992
Approved for public release; distribution is unlimited.
A~
'lii MAY 24 1993~
FLIGHT DYNAMICS DIRECTORATEWRIGHT LABORATORYAIR FORCE SYSTEMS COMMANDWRIGHT-PATTERSON AIR FORCE BASE, OHIO 45433-7913
93-11487
•8 * "> -th)r•693 I ~I4 III
NOTICE
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This technical report has been reviawed and is approved for publica-tion.
DATTA GAITONDE W. PHILLIP WEBSTERVisiting Scientist Techncial Manager
CFD Research Section
DAVID R. SELEGANChiefAeromechanics Division
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~.sbe.nI t.'itt l;th" ,d M,,eeded W~d OMPletit. I Adl -0,ew,n, th. 041r 6- It " I e.-d -A1-1flt re.Iaid'A IPtS triden eltimate Ur 4.0 ,lht e, .-4 1 ifth..
otett3 i s t,,n A,tn id-nq s snqqesltuo for redu.c~inq thlt lidnrdi to WahnweId it,es,0, 1atoue fo' t,,i o OI)etaIontf and keb(,'1s. 1d ;11 Sette(%n2OaVst t4qhw•.,Iut 1204 flhnt)sn, VA M W) 4 00 4.nd to ttw Officeiof MAnq. fMCn' ,f d t Je P41e wnOlK R•dwlion Pro t 1l0 104 0188),W 4 IV&, ht 10 . 0( )so I
1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
I Jan 93I Final Report. Dec 91 to Aui 924. TITLE AND SUBTITLE S. FUNDING NUMBERS
Calculations on Unsteady Type IV Interaction at Mach 8 F33615-90-C-2089PE- 62203PR-2307
6. AUTHOR(S) TA-N6WU-7
Datta Gaitonde
[7. PERFORMING ORGANIZATION NAME(S) AND AODRESS(ES) 8. PERFORMING ORGANIZATION
Universal Energy Systems, Inc. REPORT NUMBER
14401 Dayton-Xenia Road'Dayton OH 45432
9. SPONSORING/MONITORING AGENCY NAME(S) AND AODRESS(ES) 10. SPONSORING/ MONITORING
2light Dynamics Directorate AGENCY REPORT NUMBER1Wright Laboratory (WL/FIM JOSEPH SHANG 513-255-6156)Air Force Materiel Command WL-TR-93-3002Wright-Patterson AFB OH 45433-7913
11. SUPPLEMENT.ARY NOTES
12a. DISTRIBUTION /AVAILABILITY STATEMENT 1 2b. DISTRIBUTION CODE
Approved for public release;distribution is unlimited.
13. ABSTRACT (Ma~xmum 200 words)
The viscous unsteady flow due to a Type IV shock-on-shock interaction at Mach 8 isexamined with an implicit flux-split scheme. A sequence of grids is utilized toperform a grid resolution study and the results are compared with experimentalvalues. The formulation is finite-volume with higher order accuracy obtained withthe MUSCL approach in conjunction with a limiter to prevent oscillations. Viscousterms are centrally differenced. The flow exhibits a limit cycle in aggregatequantities which can be related to the details for the flow structure. Thesupersonic jet bounded by the two shear layers displays large scale movement at a
;dominant frequency of 32 kHz. This is accompanied by variation in the angle between!the terminating jet bow shock and the surface tangent. Unsteady separation regionsare observed both above and below the point of jet impingement.
14, SUBJECT TERMS IS. NUMBER OF PAGES
Hypersonic, high-speed, Type IV Interaction Flux-splitting, 41upwinding, implicit, unsteady 16. PRICE CODE
1. SECURITY CLASSIFICATION 18 SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF REPORT OF THIS PAGE OF ABSTRACT
ýUnclassified IUnclassified 1Unclassified Unclassified _NSN 7540 (11 280 5500 Standard Form 298 (Rs~v 2 89)
i91 -i~,e,) b it AN%'uI da 114 'PftttO
Contents
List of Figures ......... ..................................... iv
List of Tables ......... ..................................... v
Acknowledgement s .......................... ..
1 Introduction ......... .................................... 1
2 Theoretical Model ........................................ 6
3 Boundary Conditions and Numerical Details ..................... 9
4 Results ........... ........................................ 10
5 Conclusions ......... ..................................... 26
6 References .......... ...................................... 27
Nomenclature ............................................. 31
AacesB!0on For
!STIS ,G ADT~TAB
Un#vl(,r flced 5
IDiqtributi~on/AvaiibilitY Co4US
Dist Avdoc-al
List of Figures
I Schematic of Type IV interference pattern ........................ I
2 Structure of grid employed ........ ........................... I I
3 Mach contours on coarse mesh ....... ......................... 12
4 Log residuals on Grid 2 Type IV interaction.. ...................... I
5 Comparison of implicit and explicit algorithm time histories on Grid 2 -
Type IV interference ........ .............................. 17
6 Time history of Type IV interaction on Grid 3 ..................... IS
7 Representative maximum surface loading cycles - Grid 3 Type IV interact iol 20
8 Mach contours at various points in limit cycle - Grid :3 Type IV interact liou 21
9 Surface loading at various points ii, limit cycle - Grid 3 Type IV interat ioll 22
iv
List of Tables
1 Shock Interference Grid Details . ...................... . 13
Acknowledgements
Computations were performed on computers at WPAFB and the NAS program. The
support and assistance of the staff at both locations are gratefully acknowledged.
vi
1. Introduction
A pacing item in the development of hypersonic vehicle design is the capability to accurately
predict the heating on cowl-lips due to shock-on-shock interactions [1]. The forebodies of
vehicles proposed for high speeds typically act as a compression system which facilitates
optimum mass flow through the inlet. Depending upon the flight condition, the resultant
system of forebody shocks may interact with the bow shock due to the cowl-lip of the
inlet. The resulting interaction, often denoted "interference pattern," can lead to severe
amplification of pressure and heat transfer on the cowl-lip. This may have catastrophic
consequences for the vehicle unless accounted for in the design.
Shock-on-shock interactions in the presentce of a cowl-lip can result in a range of patterns
depending primarily on the location of the intersection between the impinging and cowl
shocks. Edney [2] classified these distinct interactions into six different categories: Types I
to VI. The highest surface loading occurs in the Type IV interaction shown schematically in
Figure 1. This pattern is obtained when the impinging shock intersects the nearly normal
portion of the cowl bow shock. Two triple points are formed from which emanate two
shear layers. The flow between these passes through a series of weak oblique shocks and
expansion fans until the supersonic jet so formed strikes the surface of cylinder (or cowl-lip)
after the formation of a terminating jet bow shock. The flow then expands on either side of
the stagnation point. The jet impingement process results in the maximum augmentation
of peak heat transfer an(I surface pressure.
Several studies have investigated shock-on-shock interactions at a range of free stream
Mach numbers. Following the efforts of Edney, the problem has been addressed with
both experimental [3-5] and theoretical efforts. Among the latter are the semi-emnpirical
methods of [lains et al. [6], Morris et al. [7] and the graphical methods of Crawford [8] and
Bramlette [9].
Oistofted Bow Shock
8.03
Shear Layerf
M M5.46 Sher a N xSSp-7.125 P.. ii rNm~hcS9-w3.33P..
Flow AngI = 12.5 '
Moo = 8.03Cylinder Radius = 1.5in.
Too 200.8RTwall 530R
Impinging shock angle - 18.10
Figure 1: Schematic of Type IV interference pattern
2
( om1pitatiouald solutions with the Navier-Stokes equationis have b~een reportedi by several
researchers, including Tannehill Pt al. [101, White et al. [III, Klopfer et al. fi 21, Peery 0t
al. [ :3j, Mloon et al. [ 141 and 'Uhareja et al. [1.51. ThIese studies employ a range of numerical
algorithmns inll~Uding finite-difference and finite-volumre imiplemen~tationis of centered and
characteristic-based schemes. In addition to providing a quantitative assessment of the
amp~lification in surface loading, these solutions have advanced the understanding of the
characteristics of s hock -on -shock interactions. The resulting pattern, for example, has been
shown to be very sensitive to the. location of the point of intersection of the impinging and
bow shocks.
One of the few unisteady calculations to dlate was reported by Klopfer et al. who
examined the effect of varying the impinging shock location dynamiica~lly, as would be
expected inI a flight vehicle. As the Impinging shock is lmovedl from one extreme position to
the other. all of the 1)atteriis of ELdney were ýsuccessively obtained. Nevertheless. a common
undherlyinlg assumption in each of the above calculations hias been that for a given locationl
of I lie Impinging shock, i.e., for z. given interaction pattern, the flow is steady.
However, there is reason to believe that, sonie Type IV interactions are minsteadlv. A
concrete example is presented in Figure I which focuses onl the Mach 8 Type IV interactionl
examiined in this research. Time impinging shock is created upstreamn by a wedge of 130 angle
resulting in lpost-ilnpingiflg shock flow of Mach 5.25. The cowl-li) is adequately simiulatedI
by a. cylindler. This configuration was investigated ex perimnent ally by Wieting et al. [51 and(
has b~een exam-ifled in several computational studies in recent years'.
Although thne flow was assumed to be steady initially, Holden 161] suggests that, unsteadi-
tiess lin the freqluency range 13etweeni 30 to I101,11z, is evi&-nt. in surfle- loading experimental
dlata. Indleed, several c:ompIutational stud(ies have noted a difficulty in obtaining convergedi
resinlt~s for Type IV interactions nlot, only lindele t lle above condiutions [12., 1,3] but, at, higher
Nlacli niumbiers [1 7] as well. Hlowever, no linikage was p~roposedl between the unsteadiness III
3
aggre~ate quantities atad coherent flow structures.
t'he first attempt at analyzing the details of the unsteadly flow was performed by Kroll
et al. 1181. Their research examined the interaction under inviscid conditions witij a range
of upwind and centered schemes. [In a detailed study that included] several tinle-acuiuale
0 integration methIods and] degrees of mesh resolution, they simnulatedl the ilnst~eadlitte~s- of t he
interaction and relatedl it to the coherent. flow feat ures. They concluded that the 1nust 'ad v
limit cycle consists of an oscillation of the terminating jet bow shiock about a rueaii po-ol ion.
This is accompanied by attendant effects inI the remnainder of tie flow. Further 'Iisciis"'ion
of their results is presented in Chapter 4.
This inviscid analysis, while useful in characterizing the unsteadliness, fails to provide
critically important information about, thermal loading. Furt her, visc-ous effects ate ant h'i-
patedl to be significant in the stagnation region in the close vicinity of thbe leraiataI~l ing jet
bow shock. The impact of viscosity on the nature of the unsteadiness Is ntot iaitnediately
evident. It is the p~rincipal purpose of this research to address these questions through
numerical simulation.
In recent years, significant advances have been attained in thle developmient of nimitnerical
schemes for the Euler equations. Unlike the centered schemes Of Refs. [1!) 211, thec newer
schemes use upwind methods to obviate the riced for explicit dlamping terms. Th'lese newer
methods may be classified into the means hy whiich higher order accuracy is attained
MUSCL (Monotonic U pwind-( enteredl Schieme for Conservation Laws) or noaa-MUTS( .
Examples of the former are the flux-vector split m~ethodls of Steger andl Wkarming [221 and
of van Leer (231J and thre flux-difference splhit method of Roe [211. Methods obtaining higher
ordler accuracy with iron- MUS(L m ethods i ncludle tire "upjwind" a~nd "symmetric" 'FVD
(Total Variation lDit'inishing) methods of Yee [251 and Ilarten and( Yee [261, resp~ectively.
Althouagh preliminary calc'ulations with the van Leer and Rioe schemes a-re rep~orted in)
this work, the principal method employed is that, of MacCormaack and Candler [271 denoted
4
as thte MC meithiod. A bIief (lesci-ipt.ion is pi-ovided ill ( haptvir 2 fol Iowce I by at no oi
t he 1)01! idat'y cond~itionIs and other. 1numrclical detad s ( haptei. :). J'l nail. the zesu its ar~e
examtiled ill Chapter I whilch Incliides all inve.~igattloll of the iffc~t of Inesh re-sollutiol ats
WeIll as of dic n Ill(-- aCCur acy of the algortitli.
2. Theoretical Model
The 2-D full Navier Stokes equations arc solved in strong conservation form:
Op aF'+ -j-T + Jyj = 0 (2.1)
The ý coordinate lines wrap around the cylinder while the q/ lines are body-normal 0h1 5s
forming an orthogonal mesh. In Equation 2.1, U is the solution vector of conserved variables
(U = {p, pu, pv, pe)/.J) while the flux vectors, F and C6 include the full viscous and inviscid
terms. The perfect. gas assumption is invoked, the molecular ýiscosity is approximated by
Sutherland's law and the molecular Prandti number Pr is assumed to be constant (0.73).
Equation 2.1 may be interpreted as describing the balance of mass. momentum and
energy fluxes over an arbitrary control vowbme. With first-order Euler inhplicit tine (ds-
cretization and linearization of the fluxes in time, the discretized e(Iuation may be written
in the fnrm:
{NUMERICS} bU = PHYSICS (2.2)
where PHYSICS represents the residual, NUMERICS contains the driving terms and bLt
denotes the change in the solution vector at each time step. The full Navier-Stokes equat ions
are utilized in computing the residual with the appropriate upwind model for the inviscid
terms. Viscous flux terms are centrally evaluated.
For simplicity, two assumptions are employed in evaluating the NUMERICS portion of
Equation 2.2. First, the spatial differences in flux Jacobians are calculated with the first
order spatially ac-urate Steger Warming method. This enhances the diagonal dominance
property of Equation 2.2 [281 leading to a roblust solution procedure as described below.
Secondly, the thin layer approximation is employed for the viscous Jacobians resultimig. in
a significant reduction of computational requirements.
Time integration is achieved with a residual driven line Gauss-Seidel relaxat ion schenme
G
as described in Ref. [29]. With this choice, E"quation 2.2 represents a block tridiagonal
system. The method is unconditionally stable for model diffusion and co,,vection equations
and is also known to be relatively insensitive to Lhe choice of time increment per iteration [30]. The formal time-accuracy (first order) of the resultant algorithm is not affected
by the simplifications described above. Nevertheless, the use of inconsistent differencing
needs f'irther study particularly since the NUMERICS portion employs first order spatial
derivatives. A detailed analysis of the time accuracy of the method is deferred to Chapter 4.
A brief description of the spatial scheme is presented with reference to the evaluation
of the Cartesian flux (G)at a j + 1- surface of the cell centered finite volume formulation.Trhe extension to general coor(linetcs is straightforward [31] as is the implementation of
the viscous terms which are centered. At each interface, the state of the flow is describe(d
by two vectors of conserved variables, UL and UR on either side of the interface. These
are obtained by upwind extrapolation to the cell surface with the MIISCL approach of
van Leer [32J in conjunction with a limiter. With WV denoting the vector of extrapolated
(plantities:wL. = w,+± WR K
where A introduces the limiter function, f:
-Aj+• f fA+L, Aj- 1); A T+1 Wj+- Wj+1" (2..,
ln he present work the primitive variables (W = {p, u, v,p}) are extrapolated and U
is then( reconstructed. The limiter f, preserves nonotonicity within the solution. Many
limiter functions have been proposed in the literature. The calculations in this work emnploy
the minmod limiter [33] chosen for its popularity and robustness:
f (x,y) = mininod(x,y) = sgn(x) . max {0,min[j.rj,y sgn (.)]} (2.5)
Following Steger et al. [22], the first order homogeneous hyperbolic property of the
flux .Jacobian B, (B = IG/OLT, G = BU), is utilized to split the Iluxes into positive and
negative components:
B = Q-AQ = Q-'(A+ + A_)Q = D+ +/3_ (2.6)
where A is a diagonal matrix consisting of the eigenvalties of B, A = diag {v, I + c, v - r).
A+ and A- denote the splitting of the eigenvalues into positive and negative comnponenits.
At a face j + 1/2, therefore, the original Steger Warming (abbreviated SW) flux (G.5w)
may be written as:
Gswj+L = BLUL + B"UR (2.7)
The inviscid flux formula of MacCormack and Candler [27] (GtIc) reads:
GMC4 + = B+: UL + B_ UR (2.8)
where UL and UR are evaluated with the MNUSCL approximation (Eqns 2.:3 and 2.4). The
essential difference between the SW and the MC schemes is in the manner in which the
flux Jacobians (B) are evaluated. While SW employs upwinding for the Jacobians, MC is
centered. Additional details of the MC method may be found in Ref. [:34].
The MC scheme was developed for application only in the boundary layers, and in fact.
leads to instability when applied in regions of discontinuities such as shock waves. As a
result, it is necessary to revert to the SW scheme near shocks. This is achieved in a smooth
manner by defining a pressure based switch, xi:
1XI = (2.9)1 + PR x PR
PR = IPij+I - PiI (2.10)Pnin(pij+4 ,pij)
and defining the flux at the j + 1/2 face a-s:
Gj+L = xGMCu + (1 - xl)Gsw (2.11)
8
3. Boundary Conditions and Numerical Details
The boundaries may be categorized into the following. At an inflow boundary, the flow
vector {p, pit, pv, pe} is specified according to the known freestream values. At solid bound-
aries, the velocity vector and the normal pressure gradient are assumed to be zero and at a
constant surface temperature. At outflow boundaries, the flow is assumed predominantly
supersonic and the zero gradient extrapolation condition is applied. The boundary condi-
tions for the implicit portion of the algorithm are (lescribed in Ref. [35].
The convergence or the absence of it for unsteady calculations is determined by moni-
toring several quantities. The global residual, defined as:
IIG.R.I- 1 IL -4( (Rk)i,j 2 (.1)(IL)(JL) j1j1k1U,,
is a measure representative of the overall solution. In Equation 3.1, k denotes the kth
component of U on an IL x JL computational mesh and (Rk),X, is the change in the flow
solution, Rk = AUk at i,j and U, = {p, pu, pu, pc•I.. For plotting purposes, these values
are normalized by the value of IIG.R.II obtained after the first iteration. Convergence is
assumed when IIG.R.II drops S or more orders of magnitude. After this, the surface pressure
and heat transfer are monitored over several two characteristic times, T, = L/U,,, where
[U,, is the freestream velocity and L is a macroscopic length scale - the diameter of the
cylinder or the length from the leading edge to the corner. In addition, the integrated
root inean square (RMS) pressure and heat transfer values over the entire surface are also
monitored. With P and Q denoting pressure and heat transfer respectively:
(P)RMS = -Ei (3.2)
(Q)R s = .. )IL
(QRM = Zf(Qaij=surfaCe) (3.3)
9
4. Results
The parameters for this flow are chosen after consideration of available experimental data
from Wieting et at. [5]: Mo, = 8.03, Cylinder Radius = 1.Sin., T, = 200.8R and Twall =
530R. The impinging shock angle is 18.1° corresponding to a post-shock Mach number of
5.25. The experimental results were reanalyzed and published by Thareja et al. [15] who
also performed a computational study in which they varied tile imping:ng shock location
to obtain the best fit with experimental data. Frorn their work, we choose the reh'ren ce
point for the impinging shock as (-3.5/l.5R, -0.522/1.5R) where R is tile radius and the
point (0,0) corresponds to the cylinder center. This is denoted as Case SO in Ref. 115].
Figure 2 illustrates the general features of the mesh employed. The inflow compluta-
tional boundary is assumed to be a sequence of piecewise smooth spirals bounding the
domain of interest. In the radial direction the grids are stretched algebraically. Three grids
are employed in this study with the coarser meshes (Grids 1 and 2) being extracted sys-
tematically from the finest mesh (Grid 3) in a multigrid fashion. The salient characteristics
of each grid are presented in Table 1.
In the coarse mesh calculation thie basic features of the flow structure are reproducved
(Figure 3). However, the shock system exhibits significant staircasing and the details
of the supersonic jet and the bounding shear layers are not discernible. One interesting
aspect of the calculation is the dependence of the final solution upon the time step size
utilized. At low time steps, an unsteadiness is evident in the global residual criterion and,
to a lesser extent, in the aggregate surface quantities (RMS pressure and heat. transfer).
On the other hand, at larger time step sizes, the solution converges to machine accuracy.
Preliminary calculations with both the van Leer and Roe methods exhibit unsteady limit
cycles in each of the monitored quantities at all achievable time-step sizes. The employment
of characteristic boundary conditions on the outflow boundaries has no effect upon these
10
0.25-
0.05-
-0.15-
-0.35 -11
-0.4 -0.3 -0.2 -0.1 0.0x
Figure 2: Structure of grid employed
I II
Figure 3: Mach contours on coarse mesh
12
Table 1: Shock Interference (rid Details
Grid IL x JL RccJ,,,,,, Recima. Rej.cl A19 1i, A81m., A01.. cly,,
1 49 x 24 5.6 17.9 10.4 1.8 4.0 3.1 2.0
2 99 x 49 2.2 7.5 4.3 0.9 2.0 1.5 1.4
3 199 x 99 1.0 3.4 2.0_ 0.45 1.0 0.8 1.2
Legend: IL - Points in • direction JL - Points in ?I direction
Re, - Surface mesh Reynolds number AO - Angular spacing (deg)
c - Stretch factor at surface
Subscripts: av - average rin - minimum
max - miaximium
limit cycles which are described in more detail below.
The apparent contradiction in the asymptotic behavior among the three algorithms
(M(C. van Leer and Roe) is examined further with reference to the numerical method,
For problems with steady-state solutions, the code is typically operated at the highest
achievable inviscid CFL number. The MC scheme is most robust in this regard. Operational
CFL numbers between 1000 - 5000 are achieved with the higher number pertaining to the
coarser meshes. In contrast, achievable CFL numbers with Roe's scheme are an order of
magnitude lower while the van Leer scheme is the least robust (CFL < 25). It is known
that at high time-step sizes and with a fixed number of sweeps, the Gauss-Seidel algorithm
exhibits maximal damping [30] indicating the possibility of false steady results with higher
At values. On the other hand, Yee et al. [36] state that "discrete maps" of nonlinear
equations can produce solutions which do not satisfy the original continuous system. Such
solutions can manifest themselves as spurious, unsteady cycles of any period, and may be
1:3
obtained at time step sizes above or sometimes even below the linearized stability limit.
They suggest that unsteady asymptotic behavior be examined in the limit n --+ 0, ,At --+ 0
where it is the number of interations.
The time accuracy of the present code is first investigated on the intermediate mesh. The
discretization utilized is based on the Euler implicit formula and is, therefore, formally first-
order time accurate. The solution procedure, Gauss-Seidel line relaxation, may be viewed
as time accurate within the framework of the sub-iteration strategy [28]. Neverl lihlhss. I lie
use of inconsistent differencing (through the use of SAeger WaxriiltA .Jacol)iaus) on the left
hand NUMERICS portion is relevant to this issue and a detailed examination is perfornled
in the limit At -. 0 by a combination of increased umbl)er of sweeps per iteration (from two
to four) and a reduction in At. At each time-step, therefore, the equations are solved to
within a certain tolerance. A consistent strategy of reducing At is followed until the results.
monitored with the above mentioned aggregate and local quantities, are independent of At.
The assumption that the limit cycle is independent of the initial conditions is implied and
tested by starting the soh. *-n from a converged blunt body shock and separately from a
uniform flow condition. The time-step size independent limit cycles in both instances are
identical.
The process outlined above is discussed with reference to Grid 2. A limit cycle is first
obtained with a relatively high but constant At, corresponding to a CFL number varying
slightly about 4000. The time-step is successively halved and the solution marched until
a new limit cycle is reached. This process is halted when halving At has no perceptible
influence upon the limit cycle as reflected on several aggregate and local quantities. The
results are shown in Figure 4(a) for the quantity log(JIGRII) of Equation 3.1. The extreme
values in the residual exhibit a four-period type of oscillation. Although the profiles of the
other quantities monitored are different, the influence of time-step reduction is similar. The
initial reductions in At result in an increase in amplitudeof oscillation. Below At = 0.003T,.
14
10-;
ImplicitCFL
250 125 62.51063
0 (a).4
3104- '
380.0390.0 400.0 410.0 420.0 430.0 440.0
T/Tc
Explicit
-I 10".Co
%Iwo
301FS10' -
0.0 20.0 40.0 60.0
T/Tc
Figure 4: Log residuals on (rid 2 Type IV interaction
corresponding to a CFL number of about 125, no further influence of redit'/.ion is observe(d.
The implicit solution is compared with results from an explicit second order time and space
accurate method - a two-step predictor corrector algorithm (tieun's scheme). This method
yields time-step size independent results at the linearized stability limit (CFL - 0.9). The
entire log(jIGRIJ) history of the explicit algorithm is shown in Figure 4(b).
Results from the explicit and implicit algorithm are compared in Figure 5 for several
quantities monitored over one cycle spanning the dominant frequency. In these figures. t'.fe
profiles from the explicit calculation are translated along the time axis until they' match the
implicit time development. The displayed quantities a.-e the aggregate quantities: global
residual, the instantaneous RMS surface pressure and heat transfer over the entire surface
and the local quantities: maximum surface pressure and heat transfer which occur in
the vicinity of the location of the impinging jet. It is clear that the error in aniplitiide
and frequency between the implicit and explicit methods is negligible for all quantities
monitored. Indeed, the largest discrepancy occurs in the maximum heat transfer rate on the
surface where the relative error is only 0.1%. Although the unsteadiness in this cOniputal ion
is evident in the aggregate quantities presented, the variation of local quantities is niot
pronounced. For example, the difference between the peak and trough in the inaximunu
heat transfer cycle is only 2% of the mean while that for pressure is 1.255%. The detailed
flow features reveal only a modest degree of unsteadiness in the overall flow structure.
The unsteady features are more apparent in results obtained on Grid 3. These re-
stilts were obtained at At = O.00lTd. Figure 6 displays the computed limit cycles of the
log(JjGRIJ) and maximum pressure and maximum heat transfer, respectively. Thl'e basic
four-period type oscillation similar to that observed on Grid 2 is discernible although the
regularity does not persist. The range of values obtained by the maxinium heat. transfer
and pressure is much larger, however (65% and 30% about the niean, respectively). A
spectral analysis of these limit cycles reveals a dominant frequency of about :12klz. In
16
4.0436.6 4370 437. 438
RUS Surface Preemrs RMS Surace Heat 1fansfs -0'
100.10
22.9
22.6 x3tEp
0.61 gym5
C
S7.700z- 60 ExI- xp f
X 7.676
17
I0"
I
T fTr (a)30.0 35.0 40.0 45.0 50.0
13.0-
12.0
6.,0 Si 40.0 OLD. 66.0
14.0
a.*is 10.0
*.0
14.0a- to.o @•
T/T 5 (C)4,0 T.1.0 3iO 40.0 4i.0 60.0
Figure'Fillun' hisLtoy of Type IV interaction on (Crid 3l
18
(0111)arlison, the exp(erimentally observed frequenciles ot Type IV IV ilteractiolls are in ithe
range of 3 to 10kilz [16]. An estimate of the frequency Supportable by Mesh 3 may be
based on a wavelength of 2'Ax and a sonic wave speed in the stagnation region. (Choo.irug
Ax as the grid spacing in the circumferential direction, the resolvable frequency (1080k lii)
is much higher than that observed in the present physical phenomenon. The numb)er of
time-steps within each cycle is about 800.
A small portion of the residual cycle spanning two consecutive crests of maxinmn
pressure representing the dominant frequency is isolated for further study (Figure 7(a)).
Tie maximum heat transfer cyrle is shown in Figure 7(b). The time between thue two
pressure peaks is 0.75T, corresponding to a frequency of 304Hz, The maximum he.at
transfer is similar to the peak pressure with a small leading phase.
Fuive points denoted A,B,C,D and E are identified oil the pressure profile between i lie
two successive peaks. '[he Mach contours at these points are shown il Figure S. Thle
distorted bow shock as well as the impinging shock are captured within at most two zons.
The shear layers emanating from the two triple points are also well resolved. However. sourue
portions of the distorted bow shock exhibit the "zig-zag" phenomenon which can result in
a degradation of accuracy [37].
The most significant difference in the Mach patterns appears to be the angle made bY
thi terminating jet shock with the surface of the cylinder. This shock oscillates about two
positions. At one extreme (location A) it forms an acute angle with the surface tangent of
I lit cyliuoder at tile sta.gnat iol point. At the other extreme (location C), the terinaat ing jet
shock is parallel to the cyliuder surface tangent. This shock movement is accompanied Iw
chanlges in the angle with which the supersonic jet between the two shear layers impliiuges
lijiOli the body suilrface.
Th ucsit'r•,ace pressiuire amid heat trans, fier Cespol ' (llditig to tih(le points A throiugh E ate
exhibited ili Figuire 9. The pr,.. 're is orilrillaized bY t he slagiat ioll pl-'essilrC ohbtailied wit hi
19.
Surface pressure13.0
E
WA A
'0 .0
C0.0 4.1
TI C (a)4&68.0 8 4* 48.6 4.2 4U
Heat transfer
d"l0,0
TIT, (b)4.0 1 1 T , (
4L. 48 48* 46 4*4 *
Figure 7: Representative maxiniumn surface loading cycles - (rid 1 Type JVinteraction
20
-D
o C C- 0 0 -
iO
U
Ub-U'% U
Ih-
o Q C C C 0C U is ________________
CC
a is a 0 0 8 is 0C C 0 C
2!
15.0
12.5 Expt
A10.0 B
0s ...... _7.5 .. C
5.0~a. D
15.0
2.5
0.00
• Expt
0.0 0A0
10.0- 0B
7.5 0"5.0-
2.5- Ept• (b)
0-0 ,•----••
10.0 0 B
-50.0 -35.0 -20.0 -5.0 10.00
Figure 9: Surface loading at varioits points in limit cycle - (rid 3 Type IVinteraction
"22
Ilie Rayleigh formula. For heat transfer normalization, we follow the approach of Thareja
et al. [15]. Although the experimental noninterfering stagnation heat transfer value is
6 t.7Btu/ft 2 - scc, they argue that since the theoretical value for peak heat transfer.
obtained with a VSL (Viscous Shear Layer) analysis is 41..43Btu/ft2 - sec, the difference
may be factored out by utilizing the higher (61.7) value to normalize experimental results
and the lower value (4 1.43) for computed values. The abscissa, 9. is the angle measured
with respect to the horizontal, negative values denoting the cylinder surface below the
noniinterfering stagnation streamline (Figure 8). At point A, the first peak of the pressure
cycle, the terminating jet. points upward and the corresponding mnaximum pressure occurs at
0 -18.5g. Between points A and B, the terminating jet shock assumes a position parallel
to the surface tangent and remains so until after point C. During this time, the maximum
surface pressure location moves downward to a position of 0-- -21' and the terminating
jet impinges more perpendicularly to the surface than at point A. Between points C and D.
the terminating jet shock again assumes a position similar to that at point A and remains
in this position until point E. Correspondingly the maximum surface pressure point moves
ul)wards until it reaches a maximum of - 18.2". The temporal developulent of the surface
heat trausfer is generally similar to that of the strface pressure in phase and location. After
point E, the cycle repealts in I he qualitative features though the peak values obtained differ
1" a modest. amnount. between various cycles.
Overall, the computed surface pressure overpredicts the experimental observations. T he
comipted location of t, he maximunm pressure oscillates AO - 2.5" about, a mean at 0 -,
-20'. The best agreement in magnitude of the pressure peak is obtained at point C on
the limit cycle (6% overpredicted). However, the location of this peak (-19.5") is about
-1.5 above the experimental value (-24" ). The maximum discrepancy in peak pressure is
at poilt E.
In contrast to the surface pressure, the heat transfer is underpredicted at all points in
23
the limit cycle. One explanation is based on the fact that the impinging supersonic jet is
processed through several oblique shocks and expansion falls. In this region, the effect of
the switching mechanism reverting the MC scheme to the SW scheme is not clear and iiiaN
influence the quantitative calculations in the surface quantities. Inadequate grid resoluit ion
may be another factor. The lowest value of peak heat transfer occurs at point C while
the highest value (underpredicted by 20%) occurs at point E. The locations of the heat
transfer peaks fall roughly within the experimental envelope. In the lower portion of the
cylinder, in the region between -25' and -45', both the surface pressure and heat traisfer
distributions exhibit spatial as well as temporal oscillations. 'J'h, heat transfer in this region
as well as on the upper portion of the cylinder is underpredicted. This is sinilar to the
overall observations of other researchers [12].
Further examination of the computed flowfield reveals the existence of two distinct re-
gions of unsteady separation, one below the jet impingement position (-35" to - 29")
and the other above (240 to 350). The lower separated flow is in the vicinity of thie region
where the surface pressure and heat transfer were seen to exhibit spatial oscillations. The
angular extent of this separation region, monitored with the skin friction coefficient, fluc-
tuates in time. The second separation region, above the stagnation point, is caused by a
recompression experienced by the flow as it rounds the tipper port ion of the cylinder. Thi,
compression results in a pressure rise on the surface causing boundary layer separation.
It should be noted that, in p)revious research with the El•,,ter equations [18], a recompres-
sion shock system was observed in clear detail on the upper portion of the cylinder. 'l'he
surface Mach number near the point of impingement of the compression shock was higher
than unity. This is precluded in the present research by the viscous boundary conditions.
The inviscid results of Ref. [18] for the same interaction with the Roe as well as with the
"Upwind" TVD schemes also indicate an oscillating surface pressmre profile. It that, calcu-
lation, however, the point in the cycle where the jet, terminates normal to the surface (e.g.,
24
point C above) results in the highest pressure peak in contrast to the present behavior.
On() possible explanation for this difference is based upon the influence of dissipation) upon
the strength and location of the terminating jet shock. At point C in the limit cycle, the
jet shock is nearly parallel to and farther away from the surface than at point A where the
shock is somewhat oblique to the flow. In the viscous stagnation region, the dissipation
mechanism (whether numerical or physical) not only alters the total pressure and tem[)pera-
ture recovery based upjon shock strength but may also change the information propagation
characteristics. This mechanism could introduce a phase lag between the shock system anid
the correslponding surface effects.
The source of the unsteadiness observed in the computation is not clear. The numerical
aspect of the solution has not been fully examined in the context of spatial and temporal
accuracy. The physical niechanismns acting upon the triple-point and the flow prop~erties of
the shear layers botmnding the supersonic jet are complex. In addition the shear layers are
.subject to inviscid instability and, depending upon the Reynolds number in the adjacent
shock layer, transition to turbulence can occur [16]. Further work is required to resolve
t hese issues.
25
5. Conclusions
Results for the Type IV interaction display unsteady flow behavior with all three algo-
rithms. A study of the computed unsteady flow field has been perfornwd with the M(C
scheme. A large scale movement of the supersonic impinging jet is observed with accom-
panying influences on the remainder of the flow. The dominant, frequency of the observed
phenomenon is about 32 kHz on the finest mesh. Unsteady separation regions are ob-
served both above and below the point of jet impingement. Sturface pressuires show modest
discrepancies with experiment while heat transfer rates are inderpredicted.
26
6. References
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[21 [3. Edney. Anomalous Heat Transfer and Pressure Distributions on Blunt Bodies at
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[5) A.R. Wieting and M.S. Hlolden. Experimental Study of Shock Wave Interference
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27
[9] T.T Bramlette. Simple Technique for Predicting Type III and Type IV Shock Inter-
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28
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29
[29] R. MacCormack. Current Status of Numerical Solutions of the Navier-Stokes Equa-
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[30] J.L. Thomas and R.W. Walters. Upwind Relaxation Algorithms for the Navier Stokes
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30
NomenclatureB flux Jacobian of G
Btu British Thermal IJnit
C stretch factor; local speed of sound
C, specific heat. at, collstaut volume
CFL (ourant- Friedritl-i,(vy number
total energy
internal energy
f feet
F. FP, G, flux ve'ctors
G.R. global residual
IL, JL points ill ý and 11 direction
J1 Jacobian, inverse cell volume
All Mach mmiunber
At C Mac( 1ormack and ('audler scheme
Slt Original Steger 'Warming scheme
1) pressure
p.4 i pIounds per square inch
Pr Pralidtl 1Utllbelr
PR press u re i ctioi
Q matrix diagonalizing A; heat transfer
R gas constant; Ranikine
R( Reynolds Iuimber
RAISP root miiean s(ima me surface plessum re
31
RMSQ root mean square surface heat transfer
t time
T temperature
T, characteristic time
u Cartesian velocity in x direction
U, U solution vector
v Cartesian velocity in y direction
Vf Cartesian velocity vector
VSL Viscous Shear Layer analysis
W vector of extrapolated variables
x, y Cartesian coordinates
6 cutoff value; any change
A change across interface
6 cutoff parameter
V gradient operator
"-y ratio of specific heats
A eigenvalue
A eigenvalue matrix
0 partial derivative operator
p density
0 angle
q T transformed coordinates
32
Xi pressure switch
Subscripts
J+ interface between cells j and j + It? viscous
IV wall
00 freestream conditions
Superscripts
L state at left of interface
R state at right of interface
,- positive and negative components
33
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